Lecture on Vectors and Scalars

Key Definitions

• Vectors:
  • Have both magnitude and direction.
  • Examples: Displacement, velocity, acceleration, force.
• Scalars:
  • Have magnitude only, no direction.
  • Examples: Distance, speed, energy, pressure, mass.

Difference Between Vectors and Scalars

• Non-linear paths: Example of Earth's travel:
  • Distance traveled: ~940 million km.
  • Displacement: 0 km (circular path).

Representation of Vectors

• Arrows: Indicate direction and proportional to magnitude.
• Notation: Arrow or boldface over the variable.

Vector Calculations

Vector Addition and Subtraction

• Resultant Vector: Sum or difference of two or more vectors.
• Tip-to-Tail Method:
  • Place tail of vector B at tip of vector A.
  • Arrows must maintain proportion to magnitudes.

Component Method

• Break vectors into horizontal (X) and vertical (Y) components.
• Sum of X-components and Y-components gives resultant vector components.
• Example:
  • Vectors in positive X direction: Add magnitudes.
  • Opposite Y directions: Subtract magnitudes.

Calculating Vector Components

• Given vector V, with angle θ:
  • X-component: V * cos(θ).
  • Y-component: V * sin(θ).
• Example: V = 10 m/s, θ = 30°:
  • X = 5√3 m/s.
  • Y = 5 m/s.

Finding Magnitude and Direction

• Magnitude using Pythagorean theorem:
  • V = √(x² + y²).
• Direction using:
  • θ = tan⁻¹(y/x).

Vector Multiplication

Multiplying by Scalars

• Changes magnitude, direction depends on scalar:
  • Positive scalar: Same direction.
  • Negative scalar: Opposite direction.

Multiplying by Other Vectors

• Dot Product:
  • Scalar result.
  • Formula: A · B = |A| |B| cos(θ).
• Cross Product:
  • Vector result.
  • Formula: |A × B| = |A| |B| sin(θ).
  • Direction: Use right-hand rule.

Right-Hand Rule

• Point fingers in direction of first vector.
• Curl towards second vector.
• Thumb points in direction of resultant vector.

Example Problem

• Vectors A (-3, 0, 0) and B (0, 4, 0).
• Cross product C:
  • Magnitude: 12 N·m (3 * 4 * sin(90°)).
  • Direction: Negative Z-axis.

Conclusion

• Discussed vectors and scalars.
• Next topic: Displacement and velocity.